Von Mises stresses

Assuming that all the variables follow a Normal distribution, a probabilistie model ean be ereated to determine the stress distribution for the first failure mode using the varianee equation and solving using the Finite Differenee Method (see Appendix XI). The funetion for the von Mises stress, L, on first assembly at the solenoid seetion is taken from equation 4.75 and is given by ... 

The mean value of the von Mises stress ean be approximated by substituting in the mean values of eaeh variable in equation 4.78 to give ... 

The von Mises stress, L, is then determined for various values of pre-load, E, using the above method. Equally, we eould have used Monte Carlo Simulation to determine an answer for the stress standard deviation. The answer using this approaeh is in faet o- 36 MPa over a number of trial runs. 

Fond et al. [84] developed a numerical procedure to simulate a random distribution of voids in a definite volume. These simulations are limited with respect to a minimum distance between the pores equal to their radius. The detailed mathematical procedure to realize this simulation and to calculate the stress distribution by superposition of mechanical fields is described in [173] for rubber toughened systems and in [84] for macroporous epoxies. A typical result for the simulation of a three-dimensional void distribution is shown in Fig. 40, where a cube is subjected to uniaxial tension. The presence of voids induces stress concentrations which interact and it becomes possible to calculate the appearance of plasticity based on a von Mises stress criterion. 

Fig. 42. Von Mises stress distribution for a sample containing 138 voids subjected to a deformation of 1%...

A related issue has to do with the initial wafer-level uniformity (wafer thickness, wafer warp and bow, thicknesses of thin films across the wafer surface, uniformity of stress in such thin films across the wafer) and the subsequent impact on wafer-level polish performance. Some examination has been made of the impact of wafer warp and bow on the polish performance [68], where it was found that the initial warpage can have significant impact (with the implication that reclaimed wafers may not be appropriate monitors of wafer-level polish performance). Other work has considered inherent variation due to Von Mises stress concentrations at the edge of the wafer (conceptually, a downward pressure on the wafer causes lateral stress buildup near the edge of the wafer) [64]. 

Frictional effects (shear and Von Mises stress effects)—Murthy [16] and Guo et al. [17]... 

A mathematical model was developed by Smith et al. (1998) to extract data on the elastic Young s modulus and some criteria of failure of stationary phase Baker s yeast from compression testing data. A mean Young s modulus of 150+ 15 MPa with a corresponding mean von Mises strain at failure of 0.75 + 0.08 and a mean von Mises stress at failure of 70+ 4 MPa (Smith et al., 2000b) were obtained. 

T3) are available in most FEA software packages and stresses are usually averaged by the FEA software packages to provide more accurate stress values when mapped (contoured) on to the mesh. A good first cut to the understanding of analysis results is the use of the von Mises stress (effective or equivalent stress). Fig. 10 shows the von Mises stress contour mapped to the FEA mesh in pounds/square inch (PSI). 

Figures 1 is an example for (non-dimensionalized) thermal stresses (von Mises stress) under steady-state heat conduction in a rectangular body defined over (a ,j/),0 < x < 1,0 < y < 1 subject to the Dirichlet type homogeneous boundary condition (T = 0 and Ui — 0) on the boundary with a uniform internal heat generation [5]. The thermal conductivity, elastic modulus and thermal expansion coefficient are all assumed to vary in the form of ko -f kix + k2y where fc, s are constants. For the purpose of illustrations, the values of all the material properties are taken to be unity.

Radial Stress Tangential Stress Von Mises Stress... 

The stress distributions for the different properties of the rubber sphere, for this pure hydrostatic applied stress, have been found to be unique functions of the bulk modulus, K, of the rubber (27). In other words, for a given volume fraction, the values of maximum stress for the different rubber properties fall on single curves when plotted as functions of the bulk modulus of the rubber. The relationships are shown for a 20% volume fraction of rubber in Figure 8 the values plotted are the hydrostatic stress in the rubber particle and the maximum von Mises stress in the epoxy, occurring at the interface. The results shown in Figure 8 demonstrate that the hydrostatic stress in the rubber sphere increases steadily with increasing values of K of the rubber, although the rate of increase is lower as the value of K rises. When the value of K of the rubber equals that of the epoxy annulus (i.e., 3.333 GPa), the model responds as an isotropic sphere and the stress state is pure hydrostatic tension. The maximum von Mises stress in the epoxy annulus decreases relatively... 

The variation of maximum von Mises stress in the epoxy matrix as a function of volume fraction is shown in Figure 10. The results are presented for the... 

Figure 10. Variation of the maximum von Mises stress in the epoxy matrix with the volume fraction of the rubber phase, for different values of the bulk modulus, K, of the rubber particle, and a void. The applied pure hydrostatic tension is 100 MPa.

Deformation Process. The function of the rubber in PA-rubber blends is to create stable cavities upon loading. Because of cavitation, the von Mises effective stress in the matrix strongly increases and plastic deformation is possible (I, 20). The von Mises stress in a cavitated system is a function of cavity concentration. The cavity size does not play a role in this mechanism. 

The simulation results for the von Mises stress prohle indicate that it is lowest in the nugget region and begins to increase and finally stabilize away from the center point of the workpiece (Ref 23). It is believed that further refinements to include the tool (pin) and the anvil as elements that absorb and release heat during the operahon would enhance the accuracy of the model. This model, along with the adaptive remesh ophon, leads the way to simulate the complex and dynamic phenomenon of spot FSW. 

To evaluate (7,f, the von-Mises stress due to torsion (o,) and the bending stress (cTi,) must both be calculated thus torsional stress is given by... 

Figure I 3.7 FEA of bent section of cross-ribbed PP plate a) realatively undeformed when the radius of curvature R = 4.2 m, b) some bulked ribs when R = 1.24 m. The rib and plate thickness is 2mm. Cotours of von Mises stress (MPa).

Figure I 3.12 Predicted torsion of diagonally ribbed PP beam, with contours of von Mises stress (MPa) (a) At 6 = 0.035 radm (b) 0 — 0.148 radm. ...

Figure C.3 Elastic buckling of a strut with built-in ends, and length to depth ratio 33 I, due to axial compressive forces F. Contours of von Mises stress (MPa).

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